As a short haul ( a fews days/nights at a time, no months) backpacker, I have a few knots I use every time I am on the trail. Interesting to me, that as I have grown in my knowledge, many of my knots have been replaced by others. I would be curious what others think are important knots for backpackers.
The knots I typically use include:
Constrictor knot - to secure my water bottle and throw it over a branch to hang food stuff at night.
figure eight loop - It's rare for me to use a loop while backpacking, it hasn't come up, but if my tent required line to pitch it or secure it, which many backer packers tents do, I would use this. (My tent stakes right into grommets on tabs on the tent)
trucker's hitch - any time I need an adjustable length of line, I use this instead of the taut line hitch.
I tend to use a round turn and two half hitches, alpine or overhand loop, and two half hitches when making the trucker's hitch. A fellow backpacker I know uses the rolling hitch to achieve an adjustable line.
Ian's knot - for shoelaces
Reef knot - to tie bandana around my neck or head.
To go back to the idea of prepared knots before actually going on a trip as mentioned in other posts to go sailing, etc. I use simple lanyards made of paracord, which is the only cord I take with me backpacking, and use:
Lanyard knot to form a decorative loop to hold: compass, whistle, and lighter. The use of this is totally me wanting something fancy b/c I can't help myself. Many light weight backpackers would chastise me for taking up the extra cord and therefore space and weight...I'm talking about guys who strip out the core of the paracord to save on weight.
fisherman's bend to secure lanyard
I've placed Turk's Heads on my walking staff and Moku whipped a leather hand grip on it.
I have recently begun the practice of securing my knife to my person as well. I use a length of paracord that is hitched to a small carabiner, the carabiner clipped to a belt loop. The other end of the paracord is hitched to my knife.